Impulse noise removal by an adaptive trust-region method

  • Morteza KimiaeiEmail author
  • Farzad Rahpeymaii
Methodologies and Application


This paper suggests a two-phase scheme for the impulse noise removal. In the first phase, an adaptive median filter (AMF) identifies noise candidates created by salt-and-pepper method. In the second phase, a new trust-region phase recovers noise pixels detected by the first phase. The trust-region phase produces a new adaptive radius strategy using the projected gradient for the cases where iterations are successful and a low-memory nonmonotone spectral projected gradient method (SPG) where iterations are unsuccessful. We solve the trust-region subproblem without bound constraints by the dogleg strategy (DG) using the structure of the compact limited memory Broyden–Fletcher–Goldfarb–Shanno technique. The solution is then projected into the boundary region. Numerical results are given to illustrate the efficiency of the new approach for the impulse noise removal.


Constrained optimization Image processing Trust-region framework Projected gradient strategy Adaptive radius strategy Convergence theory 



The authors are grateful to anonymous referees for their valuable comments and suggestions that improve the paper. The first author acknowledges the financial support of the Doctoral Program “Vienna Graduate School on Computational Optimization” funded by Austrian Science Foundation under Project No W1260–N35.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest

Human and animal rights

This study does not contain any studies with human participants or animals performed by any of the authors.


  1. Ahookhosh M (2018) Optimal subgradient methods: computational properties for large-scale linear inverse problems. Optim Eng.
  2. Ahookhosh M, Amini K, Kimiaei M, Peyghami MR (2016) A limited memory adaptive trust-region approach for large-scale unconstrained optimization. B Iran Math Soc 42(4):819–837MathSciNetzbMATHGoogle Scholar
  3. Ahookhosh M, Amini K (2012) An efficient nonmonotone trust-region method for unconstrained optimization. Numer Algorithms 59(4):523–540MathSciNetzbMATHGoogle Scholar
  4. Ahookhosh M, Esmaeili H, Kimiaei M (2013) An effective trust-region-based approach for symmetric nonlinear systems. Int J Comput Math 90(3):671–690zbMATHGoogle Scholar
  5. Ahookhosh M, Ghaderi S (2017) On efficiency of nonmonotone Armijo-type line searches. Appl Math Model 43:170–190MathSciNetGoogle Scholar
  6. Ahookhosh M, Neumaier A (2017) An optimal subgradient algorithm for large-scale bound-constrained convex optimization. Math Methods Oper Res 86:123–147MathSciNetzbMATHGoogle Scholar
  7. Barzilai J, Borwein JM (1988) Two point step size gradient method. IMA J Numer Anal 8:141–148MathSciNetzbMATHGoogle Scholar
  8. Bertsekas DP (1982) Projected Newton method for optimization problems with simple constrains. SIAM J Control Optim 20:221–246MathSciNetzbMATHGoogle Scholar
  9. Birgin EG, Martínez JM (2001) A spectral conjugate gradient method for unconstrained optimization. Appl Math Optim 43:117–128MathSciNetzbMATHGoogle Scholar
  10. Birgin EG, Martínez JM (2002) Large-scale active-set box-constrained optimization method with spectral projected gradients. Comput Optim Appl 23:101–125MathSciNetzbMATHGoogle Scholar
  11. Birgin EG, Martínez JM, Raydan M (2000) Nonmonotone spectral projected gradient methods on convex set. SIAM J Optim 10:1196–1211MathSciNetzbMATHGoogle Scholar
  12. Birgin EG, Martínez JM, Raydan M (2003) Inexact spectral projected gradient methods on convex sets. IMA J Numer Anal 23:539–559MathSciNetzbMATHGoogle Scholar
  13. Bovik A (2000) Handbook of image and video processing. Academic Press, CambridgezbMATHGoogle Scholar
  14. Burke JV, Moré JJ, Toraldo G (1990) Convergence properties of trust region methods for linear and convex constraints. Math Program 47:305–336MathSciNetzbMATHGoogle Scholar
  15. Byrd RH, Nocedal J, Schnabel R (1994) Representation of quasi-Newton matrices and their use in limited memory methods. Math Program 63:129–156MathSciNetzbMATHGoogle Scholar
  16. Byrd RH, Lu P, Nocedal J, Zhu C (1995) A limited memory algorithm for bound constrained optimization. SIAM J Sci Comput 16:1190–1208MathSciNetzbMATHGoogle Scholar
  17. Cai JF, Chan RH, Fiore CD (2007) Minimization of a detail-preserving regularization functional for impulse noise removal. J Math Imaging Vis 27:79–91MathSciNetGoogle Scholar
  18. Cai JF, Chan RH, Morini B (2007) Minimization of an edge-preserving regularization functional by conjugate gradient type methods. Image processing based on partial differential equations. Mathematics and visualization. Springer, Berlin, pp 109–122Google Scholar
  19. Calamai PH, Moré JJ (1987) Projected gradient methods for linearly constrained problems. Math Program 39:93–116MathSciNetzbMATHGoogle Scholar
  20. Chan RH, Ho CW, Nikolova M (2005) Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization. IEEE Trans Image Process 14:1479–1485Google Scholar
  21. Chan RH, Hu C, Nikolova M (2004) An iterative procedure for removing random-valued impulse noise. IEEE Signal Proc Letters 11(12):921–924Google Scholar
  22. Chan TF, Zhou HM, Chan RH (1995) Continuation method for total variation denoising problems. In: Proceedings of the SPIE symposium advanced signal processing: algorithms, architectures, and implementations 2563:314–325Google Scholar
  23. Charbonnier P, Blanc-Féraud L, Aubert G, Barlaud M (1997) Deterministic edge-preserving regularization in computed imaging. IEEE Trans Image Process 6(2):298–311Google Scholar
  24. Chen Z, Liu Q (2011) Convergence of affine-scaling interior-point methods with line search for box constrained optimization. Numer Funct Anal Optim 32(2):155–176MathSciNetzbMATHGoogle Scholar
  25. Chen J, Mao G, Li C, Liang W, Zhang D (2018) Capacity of cooperative vehicular networks with infrastructure support: multi-user case. IEEE T Veh Technol 67(2):1546–1560Google Scholar
  26. Chen T, Wu HR (2001) Adaptive impulse detection using center-weighted median filters. IEEE Signal Process Lett 8:1–3Google Scholar
  27. Coleman TF, Li YY (1994) On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds. Math Program 67:189–224MathSciNetzbMATHGoogle Scholar
  28. Coleman TF, Li YY (1996) An interior trust region approach for nonlinear minimization subject to bounds. SIAM J Optim 6:418–445MathSciNetzbMATHGoogle Scholar
  29. Conn AR, Gould NIM, Toint PhL (1988) Testing a class of methods for solving minimization problems with simple bounded on the variables. Math Comput 50:399–430zbMATHGoogle Scholar
  30. Conn AR, Gould NIM, Toint PhL (2000) Trust region methods. SIAM, PhiladelphiazbMATHGoogle Scholar
  31. Dai YH, Yuan Y (1999) A nonlinear conjugate gradient method with a strong global convergence property. SIAM J Optim 10:177–182MathSciNetzbMATHGoogle Scholar
  32. Dennis J, Mei H (1979) An unconstrained optimization algorithm which uses function and gradient values. J Optim Theory Appl 28:455–480MathSciNetGoogle Scholar
  33. Dolan E, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91:201–213MathSciNetzbMATHGoogle Scholar
  34. Esmaeili H, Kimiaei M (2014) A new adaptive trust-region method for system of nonlinear equations. Appl Math Model 38(11–12):3003–3015MathSciNetzbMATHGoogle Scholar
  35. Esmaeili H, Kimiaei M (2015) An efficient adaptive trust-region method for systems of nonlinear equations. Int J Comput Math 92:151–166MathSciNetzbMATHGoogle Scholar
  36. Esmaeili H, Kimiaei M (2015) An efficient implementation of a trust region method for box constrained optimization. J Appl Math Comput 48:495–517MathSciNetzbMATHGoogle Scholar
  37. Esmaeili H, Kimiaei M (2014) An improved adaptive trust-region method for unconstrained optimization. Math Model Anal 19(4):469–490MathSciNetzbMATHGoogle Scholar
  38. Fletcher R (1987) Practical methods of optimization, 2nd edn. A Wiley-Interscience Publication. Wiley, ChichesterzbMATHGoogle Scholar
  39. Grippo L, Lampariello F, Lucidi S (1986) A nonmonotone line search technique for Newton’s method. SIAM J Numer Anal 23:707–716MathSciNetzbMATHGoogle Scholar
  40. Hestenes MR, Stiefel EL (1952) Methods of conjugate gradients for solving linear systems. J Res Nat Bur Stand 49:409–436MathSciNetzbMATHGoogle Scholar
  41. Hwang H, Haddad RA (1995) Adaptive median filters: new algorithms and results. IEEE Trans Image Process 4:499–502Google Scholar
  42. Kaufman L (1999) Reduced storage, quasi-Newton trust region approaches to function optimization. SIAM J Optim 10(1):56–69MathSciNetzbMATHGoogle Scholar
  43. Kimiaei M, Ghaderi S (2017) A new restarting adaptive trust-region method for unconstrained optimization. J Oper Res Soc China 5(4):487–507MathSciNetzbMATHGoogle Scholar
  44. Kimiaei M, Rostami M (2016) Impulse noise removal based on new hybrid conjugate gradient approach. Kybernetika 52(5):791–823MathSciNetzbMATHGoogle Scholar
  45. Kwan HK, Cai Y (2002) Fuzzy filters for image filtering. In: Proceedings of circuits and systems, MWSCAS-2002, the 2002 45th midwest symposium 3:672–675Google Scholar
  46. Li SZ (1995) On discontinuity-adaptive smoothness priors in computer vision. IEEE Trans Pattern Anal Mach Intell 17(6):576–586Google Scholar
  47. Luo W (2006) Efficient removal of impulse noise from digital images. IEEE T Consum Electr 52(2):523–527Google Scholar
  48. Nikolova M (2004) A variational approach to remove outliers and impulse noise. J Math Imaging Vis 20(1–2):99–120MathSciNetzbMATHGoogle Scholar
  49. Nocedal J, Wright SJ (2006) Numerical optimization. Springer, NewYorkzbMATHGoogle Scholar
  50. Ma Z (2016) A novel compressive sensing method based on SVD sparse random measurement matrix in wireless sensor network. Engrg Comput 33(8):2448–2462Google Scholar
  51. Ma Z, Zhang DG, Cheng J, Hou YX (2017) Shadow detection of moving objects based on multisource information in Internet of things. J Exp Theor Artif Intell 29(3):649–661Google Scholar
  52. Moré JJ, Toraldo G (1991) On the solution of large quadratic programming problems with bound constraints. SIAM J Optim 1:93–113MathSciNetzbMATHGoogle Scholar
  53. Polyak E, Ribière G (1969) Note sur la convergence de directions conjugées, Francaise Informat Recherche Opertionelle, 3e Année. 16:35–43Google Scholar
  54. Powell MD (1970) A hybrid method for nonlinear equations. In: Rabinowitz P (ed) Numerical methods for nonlinear algebraic equations. Gordon and Breach, London, pp 87–114Google Scholar
  55. Powell MJD (1984) On the global convergence of trust region algorithms for unconstrained optimization. Math Program 29:297–303zbMATHGoogle Scholar
  56. Rockafellar RT (1970) Convex analysis. Princeton University Press, PrincetonzbMATHGoogle Scholar
  57. Shanno DF, Phua KH (1987) Matrix conditioning and nonlinear optimization. Math Program 14(1):149–160MathSciNetzbMATHGoogle Scholar
  58. Shi ZJ, Guo JH (2008) A new trust region method with adaptive radius. Comput Optim Appl 41:225–242MathSciNetzbMATHGoogle Scholar
  59. Song XD, Wang X (2015) New agent-based proactive migration method and system for big data environment (BDE). Engrg Comput 32(8):2443–2466Google Scholar
  60. Thomas SW (1975) Sequential estimation techniques for Quasi–Newton algorithms. Cornell University, IthacaGoogle Scholar
  61. Vogel CR, Oman ME (1998) Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE Trans Image Process 7:813–824MathSciNetzbMATHGoogle Scholar
  62. Wang CY, Liu Q, Yang XM (2005) Convergence properties of nonmonotone spectral projected gradient methods. J Comput Appl Math 182:51–66MathSciNetzbMATHGoogle Scholar
  63. Wang X, Song XD (2015) New medical image fusion approach with coding based on SCD in wireless sensor network. J Elestr Eng Technol 10(6):2384–2392Google Scholar
  64. Xu L, Burke JV (2007) An active set \(\ell _\infty \)-trust region algorithm for box constrained optimization. Technical report preprint, University of Washington.$DB_HTML$/2007/07/1717.html
  65. Yu G, Huanga J, Zhoub Y (2010) A descent spectral conjugate gradient method for impulse noise removal. Appl Math Lett 23:555–560MathSciNetGoogle Scholar
  66. Yu G, Qi L, Sun Y, Zhou Y (2010) Impulse noise removal by a nonmonotone adaptive gradient method. Signal Process 90:2891–2897zbMATHGoogle Scholar
  67. Zhang DG (2012) A new approach and system for attentive mobile learning based on seamless migration. Appl Intell 36(1):75–89Google Scholar
  68. Zhang DG, Ge H, Zhang T (2018) New Multi-hop Clustering Algorithm for Vehicular Ad Hoc Networks. IEEE T Intell Transp.
  69. Zhang DG, Kang XJ, Wang JH (2012) A novel image de-noising method based on spherical coordinates system. EURASIP J Adv Sig Process 110:1–10Google Scholar
  70. Zhang DG, Liang YP (2013) A kind of novel method of service-aware computing for uncertain mobile applications. Math Comput Modelling 57(3–4):344–356Google Scholar
  71. Zhang DG, Li WB, Liu S, Zhang XD (2016) Novel fusion computing method for bio-medical image of WSN based on spherical coordinate. J Vibroeng 18(1):522–538Google Scholar
  72. Zhang DG, Liu S, Zhang T, Liang Z (2017) Novel unequal clustering routing protocol considering energy balancing based on network partition & distance for mobile education. J Netw Comput Appl 88(15):1–9Google Scholar
  73. Zhang DG, Niu HL, Liu S (2017) Novel PEECR-based clustering routing approach. Soft Comput 21(24):7313–7323Google Scholar
  74. Zhang DG, Niu HL, Liu S, Ming XC (2017) Novel positioning service computing method for WSN. Wireless Pers Commun 92(4):1747–1769Google Scholar
  75. Zhang DG, Song XD, Wang X, Ma YY (2015) Extended AODV routing method based on distributed minimum transmission (DMT) for WSN. Aeu-Int J Electron C 69(1):371–381Google Scholar
  76. Zhang DG, Wang X, Song XD (2014) A novel approach to mapped correlation of ID for RFID anti-collision. IEEE T Serv Comput 7(4):741–748Google Scholar
  77. Zhang DG, Wang X, Song XD, Zhang T, Zhu YN (2015) New clustering routing method based on PECE for WSN. EURASIP J Wirel Commun 162:1–3Google Scholar
  78. Zhang DG, Zhao CP, Liang YP, Liu ZJ (2012) A new medium access control protocol based on perceived data reliability and spatial correlation in wireless sensor network. Comput Electr Eng 38(3):694–702Google Scholar
  79. Zhang DG, Li G, Zhang K (2014) An energy-balanced routing method based on forward-aware factor for Wireless Sensor Network. IEEE Trans Ind Inf 10(1):766–773Google Scholar
  80. Zhang T, Zhang J (2018) A kind of effective data aggregating method based on compressive sensing for wireless sensor network. EURASIP J Wirel Commun 159:1–15Google Scholar
  81. Zhang DG, Zhang XD (2012) Design and implementation of embedded un-interruptible power supply system (EUPSS) for web-based mobile application. Enterp Inf Syst-Uk 6(4):473–489Google Scholar
  82. Zhang XS, Zhang JL, Liao LZ (2002) An adaptive trust region method and its convergence. Sci China 45:620–631MathSciNetzbMATHGoogle Scholar
  83. Zhang DG, Zhou S, Tang YM (2018) A low duty cycle efficient MAC protocol base on self-adaption and predictive strategy. Mobile Netw Appl 23(4):828–839Google Scholar
  84. Zheng K, Zhang T (2015) A novel multicast routing method with minimum transmission for WSN of cloud computing service. Soft Comput 19(7):1817–1827Google Scholar
  85. Zheng K, Zhao D (2016) Novel quick start (QS) method for optimization of TCP. Wirel Netw 22(1):211–222Google Scholar
  86. Zhou S, Chen J, Liu S (2016) New mixed adaptive detection algorithm for moving target with big data. J Vibroeng 18(7):4705–4719Google Scholar
  87. Zhu Y (2012) A new constructing approach for a weighted topology of wireless sensor networks based on local-world theory for the Internet of Things (IOT). Comput Math Appl 64(5):1044–1055zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria
  2. 2.Department of MathematicsPayame Noor UniversityTehranIran

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